sicp-all-tasks/sicp/1_002e37

Exercise 1.37: 


 An infinite 
continued fraction is an expression of the form


  f
  
  =
  
  
    
      
        N
        1
      
      
        
          D
          1
        
        +
        
          
            N
            2
          
          
            
              D
              2
            
            +
            
              
                N
                3
              
              
                
                  D
                  3
                
                +
                …
              
            
          
        
      
    
    .
  


As an example, one can show that the infinite continued fraction expansion with
the 
  
    N
    i
  
 and the 
  
    D
    i
  
 all equal to 1 produces 
  
    1
    
      /
    
    φ
  
, where

  φ
 is the golden ratio (described in 1.2.2).  One way to
approximate an infinite continued fraction is to truncate the expansion after a
given number of terms.  Such a truncation—a so-called 
finite continued fraction
k-term
finite continued fraction—has the form


  
    
      
        N
        1
      
      
        
          D
          1
        
        +
        
          
            N
            2
          
          
            ⋱
            +
            
              
                N
                k
              
              
                D
                k
              
            
          
        
      
    
    .
  


Suppose that n and d are procedures of one argument (the term
index 
  i
) that return the 
  
    N
    i
  
 and 
  
    D
    i
  
 of the terms of the
continued fraction.  Define a procedure cont-frac such that evaluating
(cont-frac n d k) computes the value of the 
  k
-term finite continued
fraction.  Check your procedure by approximating 
  
    1
    
      /
    
    φ
  
 using


(cont-frac (lambda (i) 1.0)
           (lambda (i) 1.0)
           k)

for successive values of k.  How large must you make k in order
to get an approximation that is accurate to 4 decimal places?

 If your cont-frac procedure generates a recursive process, write one
that generates an iterative process.  If it generates an iterative process,
write one that generates a recursive process.